Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
Подождите немного. Документ загружается.
0 1 2 3 4 5
0.00
0.01
0.02
0.03
0.04
p
(τ)
τ
a = 8.14
0 1 2 3 4 5
0.00
0.02
0.04
0.06
τ
p
(τ)
a = 8.34
0 1 2 3 4 5
0.00
0.02
0.04
0.06
p
τ
(τ)
a = 8.48
a
τ
a > a
cr
= 2.839 . . .
A
a
p(τ)
a
a = 8.34
τ = 0.5
2.81 2.82 2.83 2.84 2.85 2.86
0.00
0.01
0.02
0.03
a
<Ω>
A=0.005
A=0.025
A=0.01
<Ω>
a
A = 0.025 A = 0.005
= 0.01 A
2.76 2.78 2.80 2.82 2.84 2.86
0.00
0.02
0.04
0.06
0.08
0.10
A
Ω=0.0153
a
a
A
Ω=0.0153
a
a
˙x = α[y − h(x)],
˙y = x − y + z,
˙z = −βy + F (t),
h(x) = m
1
x + 0.5(m
0
−m
1
)(|x + 1|−|x −1|)
m
0
=
−1/7, m
1
= 2/7 F (t) = A cos (Ωt)
β = 14.286
α
A ω
s
(A = 0)
α ≈ 8.8 α > 8.8
α A = 0.1
p(τ) α = 8.8,
ω = 0.03
s
τ
τ
(τ) p
α = 8.8175, ω = 0.045
s
τ
(τ) p
α
= 8.822, ω = 0.05
s
α ω
s
τ T = 2π/Ω
a
α
ω
ω
ω
s
s
s
ω
s
p(τ)
α
α = 8.8325 p(τ)
τ = 1
˙x
1
= σ(y
1
− x − 1) + γ(x
2
− x
1
),
˙y
1
= r
1
x
1
− x
1
z
1
−y
1
,
˙z
1
= x
1
y
1
− z
1
b,
˙x
2
= σ(y
2
− x
2
) + γ(x
1
− x
2
),
˙y
2
= r
2
x − 2 − x
2
z
2
− y
2
,
˙z
2
= x
2
y
2
− z
2
b.
σ = 10 r
1
= 28.8 r
2
= 28 b = 8/3
γ (γ > 0)
γ > 5
hf
1
i hf
2
i
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
γ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
<f>
1
2
γ
<f>
2
1
D
f
1
E D
f
2
E
σ = 10 r
1
= 28.8 r
2
= 28 b = 8/3
x
1
(t) x
2
(t)
γ > 5
N
SR
k
ξ
k
(t)
s(t)
x
M
(t)
x
M
(t) =
1
N
N
X
k=1
x
k
(t).
N 1
x
x
x
x
1
2
3
N
SR
SR
SR
SR
1
2
3
N
Σ
ξ
ξ
ξ
1
2
3
N
s(t)
x
M
.
.
.
ξ
x
x
x
x
1
2
3
N
SR
SR
SR
1
2
3
N
Σ
ξ
ξ
ξ
1
2
3
N
s(t)
x
M
.
.
.
ξ
SR
SR
k
s(t)
ξ
k
(t)
x
M
s(t)
x
M
(t)
k
G
k,k
(ω) k m
G
k,m
(ω)
G
M,M
(ω) G
M,M
(ω)
G
M,M
(ω) =
1
N
2
N
X
k=1
G
k,k
(ω) +
N
X
k=1
N
X
m=1
k6=m
G
k,m
(ω)
.
G
k,m
(ω) = 0
χ
k
(ω, D) D
k
G
k,k
(ω) = G
(0)
k,k
(ω, D) + |χ
k
(ω, D)|
2
G
s,s
(ω),
G
(0)
k,k
(ω, D) k
G
s,s
(ω)
G
k,m
(ω)
G
k,m
(ω) = χ
∗
k
(ω, D) χ
m
(ω, D) G
s,s
(ω),
∗
G
M,M
(ω) =
1
N
2
N
X
k=1
G
(0)
k,k
(ω, D)+
G
s,s
(ω)
N
2
N
X
k=1
N
X
m=1
χ
∗
k
(ω, D)χ
m
(ω, D).
G
s,M
(ω)
G
s,M
(ω) =
G
s,s
(ω)
N
N
X
k=1
χ
k
(ω, D).
χ(ω, D) ≡ χ
k
(ω, D)
G
(0)
x,x
(ω, D) ≡ G
(0)
k,k
(ω, D)
G
M,M
(ω) =
1
N
G
(0)
x,x
(ω, D) + |χ(ω, D)|
2
G
s,s
(ω),
G
s,M
(ω, D) = χ(ω, D) G
s,s
(ω).
N
χ(ω, D)
s(t) = n(t) + A sin(Ωt)
SNR
in
=
πA
2
G
n,n
(Ω)
,
G
n,n
(Ω)
δ
G
M,M
(ω) =
1
N
G
(0)
x,x
(ω, D) + |χ(ω, D)|
2
G
n,n
(ω) + πA
2
δ(ω −Ω)
.
SNR
out
η =
SNR
out
SNR
in
= 1 −
G
(0)
x,x
(Ω, D)
G
(0)
x,x
(Ω, D) + N |χ(Ω, D)|
2
G
n,n
(Ω)
.
1 N → ∞
|χ(Ω, D)|
η
N =
G
(0)
x,x
(Ω, D)
|χ(Ω, D)|
2
G
n,n
(Ω)
η
1 − η
.
s(t) = n(t) + A sin(Ωt)
n(t)
G
n,n
(ω) = γQ/(ω
2
+ γ
2
)
η = 1 −
Ω
2
+ γ
2
D
2
(Ω
2
+ γ
2
) + N λ
m
hx
2
i
st
Q γ
0.1
0.3
0.5
0.6
0.7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
100
200
300
400
500
600
700
800
900
1000
0.0001
0.001
0.01
0.1
1
D
N
η
η
Q = 10
−3
γ = 1.0 A = 0.1 Ω = 0.1
N =
D
2
(Ω
2
+ γ
2
)
λ
m
hx
2
i
st
Q γ
η
1 − η
,
N
η
η
D
N
N(D)
η
N η D
N → ∞
x
∞
M
(t) = A |χ(Ω, D)| cos (Ωt + ψ),
ψ
ψ = −arctan
Im χ(Ω, D)
Re χ(Ω, D)
.